Annals of Mathematics A local regularity theorem for mean curvature flow By Brian White Annals of Mathematics, 161 (2005), 1487–1519 A local regularity theorem for mean curvature flow By Brian White* Abstract This paper proves curvature bounds for mean curvature flows and other related flows in regions of spacetime where the Gaussian densities are close to 1. Introduction Let M t with 0 < t < T be a smooth one-parameter family of embed- ded manifolds, not necessarily compact, moving by mean curvature in R N . This paper proves uniform curvature b ounds in regions of spacetime where the Gaussian density ratios are close to 1. For instance (see §3.4): Theorem. There are numbers ε = ε(N ) > 0 and C = C(N ) < ∞ with the following property. If M is a smooth, proper mean curvature flow in an open subset U of the spacetime R N × R and if the Gaussian density ratios of M are bounded above by 1 + ε, then at each spacetime point X = (x, t) of M, the norm of the second fundamental form of M at X is bounded by C δ(X, U) where δ(X, U) is the infimum of X − Y  among all points Y = (y, s) ∈ U c with s ≤ t. (The terminology will be explained in §2.) Another paper [W5] extends the bounds to arbitrary mean curvature flows of integral varifolds. However, that extension seems to require Brakke’s Local Regularity Theorem [B, 6.11], the proof of which is very difficult. The results of this paper are much easier to prove, but nevertheless suffice in many interesting situations. In particular: *The research presented here was partially funded by NSF grants DMS-9803403, DMS- 0104049, DMS-0406209 and by a Guggenheim Foundation Fellowship. 1488 BRIAN WHITE (1) The theory developed here applies up to and including the time at which singularities first occur in any classical mean curvature flow. (See Theo- rem 3.5.) (2) The bounds carry over easily to any varifold flow that is a weak limit of smooth mean curvature flows. (See §7.) In particular, any smooth compact embedded hypersurface of R N is the initial surface of such a flow for 0 ≤ t < ∞ (§7.4). (3) The bounds also extend easily to any varifold flow constructed by Ilma- nen’s elliptic regularization procedure [I1]. Thus, for example, the results of this paper allow one to prove (without using Brakke’s Local Regularity Theorem) that for a nonfattening mean curvature flow in R N , the surface is almost everywhere regular at all but countably many times. (A slightly weaker partial regularity result was proved using Brakke’s Theorem by Evans and Spruck [ES4] and by Ilmanen [I1].) Similarly, the local regularity theorem here suffices (in place of Brakke’s) for the analysis in [W3], [W4] of mean curvature flow of mean convex hypersurfaces (see §7.2, §7.3, and §7.4). The proofs here are quite elementary. They are based on nothing more than the Schauder estimates for the standard heat equation in R m (see §8.2 and §8.3), and the fact that a nonmoving plane is the only entire mean curvature flow with Gaussian density ratios everwhere equal to 1. The proof of the basic theorem is also fairly short; most of this paper is devoted to generalizations and extensions. Although the key idea in the proof of the main theorem is simple, there are a number of technicalities in the execution. It may therefore be helpful to the reader to first see a simpler proof of an analogous result in which the key idea appears but without the technicalities. Such a proof (of a special case of Allard’s Regularity Theorem) is given in Section 1. Section 2 contains preliminary definitions and lemmas. The main result of the paper is proved in Section 3. In Section 4, the result is extended to surfaces moving with normal velocity equal to mean curvature plus any H¨older continuous function of position, time, and tangent plane direction. This in- cludes, for example, mean curvature flow in Riemannian manifolds (regarded as isometrically embedded in Euclidean space). In Section 5, the analogous estimates at the boundary (or “edge”) are proved for motion of manifolds- with-boundary. In Section 6, somewhat weaker estimates (namely C 1,α and W 2,p ) are proved for surfaces moving by mean curvature plus a bounded mea- surable function. This includes, for example, motion by mean curvature in the presence of smo oth obstacles. Finally, in Section 7, the regularity theory is extended to certain mean curvature flows of varifolds. This section may be read directly after Section 3, A LOCAL REGULARITY THEOREM 1489 but it has been placed at the end of the paper since it is the only section involving varifolds. Mean curvature flow has been investigated extensively in the last few decades. Three distinct approaches have been very fruitful in those investi- gations: geometric measure theory, classical PDE, and the theory of level- set or viscosity solutions. These were pioneered in [B]; [H1] and [GH]; and [ES1]–[ES4] and [CGG] (see also [OS]), respectively. Surveys emphasizing the classical PDE approach may be found in [E2] and [H3]. A rather thorough introduction to the classical approach, including some new results, may be found in [E4]. An intro duction to the geometric measure theory and viscosity approaches is included in [I1]. See [G] for a more extensive introduction to the level set approach. Some of the results in this paper were announced in [W1]. Some similar results were proved by A. Stone [St1], [St2] for the special case of hyper- surfaces with positive mean curvature under an additional hypothesis about the rate at which curvature blows up when singularities first appear. In [E1], K. Ecker proved, for the special case of two-dimensional surfaces in 3-manifolds, pointwise curvature bounds assuming certain integral curvature bounds. The monotonicity formula, which plays a major role in this paper, was discovered by G. Huisken [H2]. Ecker has recently discovered two new remarkable mono- tonicity formulas [E3], [E4, §3.18] that have most of the desirable features of Huisken’s (and that yield the same infinitesimal densities) but that, unlike Huisken’s, are local in space. 1. The main idea of the proof As mentioned above, the key idea in the proof of the main theorem is simple, but there are a number of necessary technicalities that obscure the idea. In this section, the same idea (minus the technicalities) is used to prove a special case of Allard’s Regularity Theorem [A]. The proof is followed by a brief discussion of some of the technicalities that make the rest of the paper more complicated. This section is included purely for expository reasons and may be skipped. 1.1. Theorem. Suppose that N is a compact Riemannian manifold and that ρ > 0. There exist positive numbers ε = ε(N, ρ) and C = C(N, ρ) with the following property. If M is a smooth embedded minimal submanifold of N such that θ(M, x, r) ≤ 1 + ε for all x ∈ N and r ≤ ρ, then the norm of the second fundamental form of M is everywhere bounded by C. 1490 BRIAN WHITE Here θ(M, x, r) denotes the density ratio of M in B(x, r): θ(M, x, r) = area(M ∩ B(x, r)) ω m r m , where m = dim(M) and ω m is the volume of the unit ball in R m . Proof. Suppose the result were false for some N and ρ. Then there would be a sequence ε i → 0 of positive numbers, a sequence M i of smooth minimal submanifolds of N, and a sequence x i of points in M i for which (∗) θ(M i , x, r) ≤ 1 + ε i (x ∈ N, r ≤ ρ) and for which B(M i , x i ) → ∞, where B(M, x) denotes the norm of the second fundamental form of M at x. Note that we may choose the x i to maximize B(M i , ·): max x B(M i , x) = B(M i , x i ) = Λ i → ∞. We may also assume that N is isometrically embedded in a Euclidean space E. Translate M i by −x i and dilate by Λ i to get a new manifold M  i with max B(M  i , ·) = B(M  i , 0) = 1. By an Arzela-Ascoli argument, a subsequence (which we may assume is the original sequence) of the M  i converges in C 1,α to a limit submanifold M  of E. By standard elliptic PDE, the convergence is in fact smooth, so that M  is a minimal submanifold of E and (†) max B(M  , ·) = B(M  , 0) = 1. On the other hand, (∗) implies that θ(M  , x, r) ≤ 1 for all x ∈ M  and for all r. Monotonicity implies that the only minimal surface with this property is a plane. So M  is a plane. But that contradicts (†). Complications There are several reasons why the proof of the main theorem (3.1) of this paper is more complicated than the proof above. For example: 1. It is much more useful to have a local result than a global one. Thus in Theorem 1.1, it would be better to assume not that M is compact, but rather that it is a proper submanifold of an open subset U of N . Of course then we A LOCAL REGULARITY THEOREM 1491 can no longer conclude that B(M, ·) is bounded. Instead, the assertion should become B(M, x) dist(x, U c ) ≤ 1. This localization introduces a few annoyances into the proof. For example, we would like (following the proof above) to choose a point x i ∈ M i for which B(M i , x i ) dist(x i , U c i ) is a maximum. But it is not clear that this quantity is even bounded, and even if it is bounded, the supremum may not be attained. 2. For various reasons, it is desirable to have a slightly more complicated quantity play the role that B(M, x) does above. For instance, max B(M, ·) is like the C 2 norm of a function, and as is well known, Schauder norms are much better suited to elliptic and parabolic PDE’s. Thus instead of B(M, x) we use a quantity K 2,α (M, x) which is essentially the smallest number λ > 0 such that the result of dilating M ∩B(x, 1/λ) by λ is, after a suitable rotation, contained in the graph of a function u : R m → R d−m with u 2,α ≤ 1. Here d is the dimension of the ambient Euclidean space. There is another reason for not using the norm of the second fundamantal form. Suppose we wish to weaken the hypothesis of Theorem 1.1 by requiring not that M be minimal but rather that its mean curvature be bounded. We can then no longer conclude anything about curvatures. However, we can still conclude, with essentially the same proof, that K 1,α (M, x) is bounded. 3. Spacetime (for parabolic problems) is somewhat more complicated than space (for elliptic problems). Thus for example parabolic dilations and Gaussian densities replace the more geometrically intuitive Euclidean dilations and densities. 2. Preliminaries 2.1. Spacetime. We will work in spacetime R N,1 = R N × R. Points of spacetime will be denoted by capital letters: X, Y , etc. If X = (x, t) is a point in spacetime, X denotes its parab olic norm: X = (x, t) = max{|x|, |t| 1/2 }. The norm makes spacetime into a metric space, the distance between X and Y being X−Y . Note that the distance is invariant under spacetime translations and under orthogonal motions of R N . 1492 BRIAN WHITE For λ > 0, we let D λ : R N,1 → R N,1 denote the parab olic dilation: D λ (x, t) = (λx, λ 2 t). Note that D λ X = λX. We let τ : R N,1 → R denote projection onto the time axis: τ(x, t) = t. 2.2. Regular flows. Let M be a subset of R N,1 such that M is a C 1 embedded submanifold (in the ordinary Euclidean sense) of R N+1 with dimension m + 1 (again, in the usual Euclidean sense). If the time function τ : (x, t) → t has no critical points in M, then we say that M is a fully regular flow of spatial dimension m. If a fully regular flow M is C ∞ as a submanifold of R N+1 , then we say that it is a fully smooth flow. It is sometimes convenient to allow flows that softly and suddenly vanish away. Thus if M is a fully regular (or fully smooth) flow and T ∈ (−∞, ∞], then the truncated set {X ∈ M : τ(X) ≤ T } will be called a regular (or smooth) flow. Of course if T = ∞, then the truncation has no effect. Thus every fully regular (or fully smooth) flow is also a regular (or smooth) flow. Note that if M is a regular (or smooth) flow, then for each t ∈ R, the spatial slice M(t) := {x ∈ R N : (x, t) ∈ M} is a C 1 (or smooth) m-dimensional submanifold of R N . Of course for some times t the slice may be empty. For example, suppose M is a smooth m-dimensional manifold, I is an interval of the form (a, b) or (a, b], and F : M × I → R N is a smooth map such that for each t ∈ I, the map F ( ·, t) : M → R N is an embedding. Let M be the set in spacetime traced out by F : M = {(F (x, t), t) : x ∈ M, t ∈ I}. Then M is a smooth flow. Conversely, if M is any regular (or smooth) flow and X ∈ M, then there is a spacetime neighborhood U of X and an F as above such that M ∩ U = {(F (x, t), t) : x ∈ M, t ∈ I} is the flow traced out by F. Such an F is called a local parametrization of M. If M is smooth, then by the Fundamental Existence and Uniqueness Theorem A LOCAL REGULARITY THEOREM 1493 for ODEs, we can choose F so that for all (x, t) in the domain of F , the time derivative ∂ ∂t F (x, t) is perpendicular to F (M, t) at F (x, t). 2.3. Proper flows. Suppose that M is a regular flow and that U is an open subset of spacetime. If M = M ∩ U, then we will say that M is a proper flow in U. For any regular flow M, if U is the spacetime complement of M \ M, then M is a proper flow in U. Also, if M is a proper flow in U and if U  is an open subset of U, then M ∩U  is a prop er flow in U  . 2.4. Normal velocity and mean curvature. Let M be a regular flow in R N,1 . Then for each X = (x, t) ∈ M, there is a unique vector v = v(M, X) in R N such that v is normal to M(t) at x and (v, 1) is tangent (in the ordinary Euclidean sense) to M at X. This vector is called the normal velocity of M at X. If F is a local parametrization of M, then v(M, (F (x, t), t)) =  ∂ ∂t F (x, t)  ⊥ . If M is a regular flow and X = (x, t), we let H(M, X) denote the mean curvature vector (if it exists) of M(t) at x. Of course if M is smooth, then M(t) is also smooth, so H(M, X) does exist. A regular flow M such that v(M, X) = H(M, X) for all X ∈ M is called a mean curvature flow. Note that if we parabolically dilate M by λ, then v and H get multiplied by 1/λ: v(D λ M, D λ X) = λ −1 v(M, X), H(D λ M, D λ X) = λ −1 H(M, X). Thus if M is a mean curvature flow, then so is D λ M. 2.5. The C 2,α norm of M at X. We wish to define a kind of local C 2,α norm of a smooth flow at a point X ∈ M. This norm will be denoted K 2,α (M, X). Actually the definition below makes sense for any subset M of spacetime. Let B N = B N (0, 1) and B N,1 = B N ×(−1, 1) denote the open unit balls centered at the origin in R N and in spacetime R N,1 , respectively. The graph of a function u : B m,1 → R N−m is the set graph(u) = {(x, u(x, t), t) : (x, t) ∈ B m,1 } ⊂ R N,1 . Now consider first the case X = 0 ∈ M. Suppose we can rotate M to get a new set M  for which the intersection M  ∩ B N,1 1494 BRIAN WHITE is contained in the graph of a function u : B m,1 → R N−m whose parabolic C 2,α norm is ≤ 1. (See the appendix (§7) for the definition of the parabolic H¨older norms of functions.) Then we will say that K 2,α (M, X) = K 2,α (M, 0) ≤ 1. Otherwise, K 2,α (M, 0) > 1. More generally, we let K 2,α (M, 0) = inf{λ > 0 : K 2,α (D λ M, 0) ≤ 1}. Finally, if X is any point in M, we let K 2,α (M, X) = K 2,α (M − X, 0), where M−X is the flow obtained from M by translating in spacetime by −X. If K 2,α (M, ·) is bounded on compact subsets of a regular flow M, then M is called a C 2,α flow. Remark on the definition. Suppose M is a proper, regular flow in U and X ∈ M. If we translate M by −X, dilate by λ = K 2,α (M, X), and next rotate appropriately to get a flow M  , then by definition, M  ∩ B N,1 will be contained in the graph of a function u : B m,1 → R N−m as with parabolic C 2,α norm ≤ 1. Note that if M is fully regular and if the distance from X to U c is ≥ r = 1/λ, then in fact M  ∩ B N,1 = graph(u) ∩B N,1 . Likewise, if M is regular but not necessarily fully regular, then for some T ≥ 0, M  ∩ B N,1 = graph(u) ∩B N,1 ∩ {Y : τ(Y ) ≤ T }. 2.6. Arzela-Ascoli Theorem. For i = 1, 2, 3, . . . , let M i be a proper C 2,α flow in U i . Suppose that M i → M and that U c i → U c as sets. Suppose also that the functions K 2,α (M i , ·) are uniformly bounded as i → ∞ on com- pact subsets of U. Then M  = M ∩ U is a proper C 2,α flow in U, and the convergence M i → M  is locally C 2 (parabolically). In particular, if X i ∈ M i converges to X ∈ M  , then v(M i , X i ) →v(M  , X), H(M i , X i ) →H(M  , X), and A LOCAL REGULARITY THEOREM 1495 K 2,α (M  , X) ≤ lim inf i K 2,α (M i , X i ). Furthermore, if each M i is fully regular in U i , then M  is fully regular in U . Remark on the hypotheses. Convergence of S i → S as sets means: every point in S is the limit of a sequence X i ∈ S i , and for every bounded sequence X i ∈ S i , all subsequential limits lie in S. The uniform boundedness hypothesis is equivalent to: for every sequence X i ∈ M i converging to X ∈ U, the lim sup of K 2,α (M i , X i ) is finite. Proof. Straightforward. See Section 8.1 for details. The last assertion follows from the remark above about the definition of K 2,α . Note that K 2,α (M, ·) scales like the reciprocal of distance. That is, K 2,α (D λ M, D λ X) = λ −1 K 2,α (M, X). We will also need a scale invariant version of K 2,α . Let d(X , U) = inf{X −Y  : Y ∈ U c }. Then of course d(X, U) K 2,α (M, X) is scale invariant. Definition. Suppose M is a proper smooth flow in U. Then K 2,α; U (M) = sup X∈M d(X , U) · K 2,α (M, X). Of course K 2,α; U (M) is scale-invariant. 2.7. Corollary to the Arzela-Ascoli Theorem. K 2,α; U (M  ) ≤ lim inf K 2,α; U i (M i ). 2.8. Proposition. Suppose M is a proper C 2,α flow in U. Let U 1 ⊂ U 2 ⊂ . . . be open sets such that (1) the closure of each U i is a compact subset of U, and (2) ∪ i U i = U . Then K 2,α; U i (M ∩ U i ) < ∞ for each i and K 2,α;U (M) = lim K 2,α; U i (M ∩ U i ). The proof is very easy. Note that for any U , there always exist such U i . 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Math 95 (1972), 417–491 [B] K Brakke, The Motion of a Surface by its Mean Curvature, Princeton Univ Press, Princeton, NJ, 1978 [CGG] Y G Chen, Y Giga, and S Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J Differential Geom 33 (1991), 749–786 [E1] K Ecker, On regularity for mean curvature flow of hypersurfaces, Calc Var Partial Differential Equations... Hence Θ(M, X) ≥ 1 for each X ∈ M Thus by monotonicity, Θ(M, ∞) ≥ 1 Furthermore, if Θ(M, ∞) ≤ 1, then Θ(M, ∞) = Θ(M, X) = 1 for every X ∈ M But by monotonicity (see the discussion immediately preceding 2.10), this implies that the set {Y ∈ M : τ (Y ) ≤ τ (X)} has the form (∗) Since this is true for every X ∈ M, in fact all of M must have the form (∗) 3 The fundamental theorem 3.1 Theorem For 0 < α < 1, . A local regularity theorem for mean curvature flow By Brian White Annals of Mathematics, 161 (2005), 1487–1519 A local regularity theorem for mean. [I1]. Thus, for example, the results of this paper allow one to prove (without using Brakke’s Local Regularity Theorem) that for a nonfattening mean curvature flow